So, two hours in, a couple of quests finished...

[Chokapi]

And after having depressed the 'P' key on my kb, I see a list of other quests. I double-click on one and am told that I haven't started that quest, and go see, say, Handsome Wilm. I do. He only asks if I've seen what-her-face. Alas, no, Wilm, haven't seen her. Seems I've run into a few of the people on the list and none seem to have the quest icon floating above. When I go back to the tavern, the bartender asks me to go on the same quest over.

But at what point does anyone provide me with another quest?

[Cedwin]

Most of the people with quest icons over their head only give out one quest. You'll need to go to another area to pick up more quests.

[gavagai]

On Korthos there's an order in which you need to complete quests before later ones open up.

Before you get Necromancer's Doom from Handsome Wilm, you need to complete the quest from the bartender ("The Collaborator").

And before that one, you need to complete the first three -- Storehouse Secret, Cannith Crystal, and Heyton's Rest.

All the quests culminate in the level 2 quest, Misery's Peak (opened after you finish "Redemption") -- at which you enter "Sunny side Korthos" and can go freely to Stormreach.

[7-day_Trial_Monkey]

*snip* -- at which you enter "Sunny side Korthos" and can go freely to Stormreach.

You can go freely to Stormreach anytime you want. No quests on khorthos need to be completed 1st.

Just go to the dock and pick the option to go to Stormreach. If you want, you can immediately go back to Khorthos sunnyside and continue your questing there.

[Frotz]

Sounds like you just haven't visited the graveyard in Korthos to run into Kaja. Open the map with M and you'll see where quest-givers are that you haven't visited.